1. Time-Varying Encoders for Constrained Systems - Jonathan J. Ashley, Brian H. Marcus Jungwon Lee jungwon@stanford.edu

2. EE392P Project: Time-Varying Encoders for Constrained Systems2 Outline Introduction Time-Varying Finite-State Encoders Approximate Eigenvectors Time-Varying State Splitting Sliding-Block Decoders Conclusion

3. EE392P Project: Time-Varying Encoders for Constrained Systems3 Introduction Data-to-codeword assignment Can significantly affect the complexity and performance of the code p small: reasonably good assignments by an ad hoc approach p large: hard to find a good assignment Break down the coding problem into smaller subproblems Periodically time-varying constraints

4. EE392P Project: Time-Varying Encoders for Constrained Systems4 Introduction Example 1: 16:19 -> 8:9/8:10 low-complexity & low error rate 16:19 block- decodable code 8:9/8:10 block- decodable code 8:9/8:10 (0,1)- SBD code 1 isolated channel error 2 bytes1 byte2 bytes 2-bit channel error 2 or 4 bytes1 or 2 bytes2 or 3 bytes xxxxxx

5. EE392P Project: Time-Varying Encoders for Constrained Systems5 Introduction Example 2: PRML (4,4) Set S of all the binary sequences such that  maximum run of 0’s is 4, and  maximum run of 0’s in both the odd and even interleaves is 4. G is a labeled graph whose states are the ordered pairs  i: length of the run of 0’s in the preceding interleave  j: length of the run of 0’s in the current interleave  Six of 25 possible states are not reachable. So, G has only 19 states. Two phase rate 8:8/8:9 (0,1)-SBD code Neither rate 16:17 block code nor block-decodable code PRML(4,4) is minimal to support a code at rate 16:17 – capacities of PRML(3,4) and PRML(4,3) are less than 16/17.

6. EE392P Project: Time-Varying Encoders for Constrained Systems6 Introduction Generality of time-varying encoders States in the different phases can have different lengths of input tags and output labels. States in the same phase have the same length. Not restricted to codes constructed by state- splitting Time-varying approach can be used for look- ahead or combined look-ahead/state-splitting encoders. Time-varying state splitting as an application of variable-length state splitting

7. EE392P Project: Time-Varying Encoders for Constrained Systems7 Time-Varying Encoders k-partite labeled graphs Vertices divided into k subsets called phases: States: k copies of vertex set of G Edges: uv (u,i) in V i (v,i+1) in V i+1

8. EE392P Project: Time-Varying Encoders for Constrained Systems8 Time-Varying Encoders presented by set of sequences in S, each expressed as a sequence of nonoverlapping q i -blocks Given and, a -encoder is a labeled graph s.t. a) each sequence of output labels obtained by traversing the graph belongs to S; b) each state in phase i has exactly n i outgoing edges; c) is lossless.

9. EE392P Project: Time-Varying Encoders for Constrained Systems9 Time-Varying Encoders A tagged encoder: n i -ary input labels for every state in phase i Rate encoder into S: tagged - encoder, where given Proposition 1: Let. The following are equivalent. 1) There is a rate encoder into S. 2) There is a rate encoder into S. 3)

10. EE392P Project: Time-Varying Encoders for Constrained Systems10 Time-Varying Encoders Proof of proposition 1 1) => 2): restrict to the states of phase 0 2) => 1): assume k=2 2) 3): results of the ordinary state-splitting algorithm PRML (4,4): capacity = 0.9614 > 16/17 = 0.9412 There is a rate 8:8/8:9 encoder.

11. EE392P Project: Time-Varying Encoders for Constrained Systems11 Approximate Eigenvectors k-partite matrix: adjacency matrix of a k-partite graph Adjacency matrix of : PRML (4,4): rate 8:8/8:9 encoder

12. EE392P Project: Time-Varying Encoders for Constrained Systems12 Approximate Eigenvectors : diagonal matrix indexed by the states of with diagonal entries where u is a state in phase i -approximate eigenvector: a nonnegative integer vector x such that  Proposition 2: Let be a k-partite labeled graph,, and. Then, there is an -approximate eigenvector if and only if.

13. EE392P Project: Time-Varying Encoders for Constrained Systems13 Approximate Eigenvectors Proof of proposition 2 =>  -approximate eigenvector is an ordinary - approximate eigenvector.   <=  There is an -approximate eigenvector x.  ny is -approximate eigenvector.

14. EE392P Project: Time-Varying Encoders for Constrained Systems14 Approximate Eigenvectors Modified Franaszek Algorithm PRML (4,4):

15. EE392P Project: Time-Varying Encoders for Constrained Systems15 Approximate Eigenvectors Merging state v into another state u u and v are in the same phase. u and v have the same approximate eigenvector entry. Any output label sequence generated from u can also be generated from v. PRML (4,4): has eight states. -approximate eigenvector is, where.

16. EE392P Project: Time-Varying Encoders for Constrained Systems16 Time-Varying State Splitting x-consistent partition:, where i is the phase of state u x-consistent splitting: state splitting based on such a partition

17. EE392P Project: Time-Varying Encoders for Constrained Systems17 Time-Varying State Splitting Encoder construction given and Find an -approximate eigenvector using the modified Franaszek algorithm. Do a sequence of x-consistent splittings. End with a k-partite lossless presentation s.t. the entries of the induced -approximate eigenvector are at most 1. After deleting states with vector entry 0, each state of phase i has out-degree at least. The resulting graph is an -encoder.

18. EE392P Project: Time-Varying Encoders for Constrained Systems18 Time-Varying State Splitting More merging u and v are in the same phase. Any output label sequence generated from u can also be generated from v. The weight m of u is less than the weight n of v. => m descendants of u can be merged with m of the n descendants of v. PRML (4,4): After state-splitting, 22 states in phase 0 and 30 states in phase 1 After merging, 8 states in phase 0 and 11 states in phase 1

19. EE392P Project: Time-Varying Encoders for Constrained Systems19 Sliding-Block Decoders Sliding-block decodable: begin in the same phase (m,a)-definite: begin in the same phase x -m ’/y -m e -m ’ x0’/y0x0’/y0 e0’e0’ x a ’/y a ea’ea’ x -m /y -m e -m x0/y0x0/y0 e0e0 xa/yaxa/ya eaea x 0 = x 0 ’ x -m ’/y -m e -m ’ x0’/y0x0’/y0 e0’e0’ x a ’/y a ea’ea’ x -m /y -m e -m x0/y0x0/y0 e0e0 xa/yaxa/ya eaea e 0 = e 0 ’

20. EE392P Project: Time-Varying Encoders for Constrained Systems20 Sliding-Block Decoders Finite-type constrained system G: finite memory : (m,0)-definite State splitting of an (m,a)-definite graph always yields an (m,a+1)-definite graph. Proposition 3: Let S be a finite-type constrained system. Let and. Then, there is a sliding-block decodable rate encoder into S if and only if.

21. EE392P Project: Time-Varying Encoders for Constrained Systems21 Sliding-Block Decoders Proposition 4: Then, (0,a)-sliding-block decodable. PRML (4,4): The compatibility conditions with one block of anticipation and no memory are satisfied. x0/y0x0/y0 e0e0 (u,i) x1/y1x1/y1 e1e1 xa/yaxa/ya eaea x0’/y0x0’/y0 e0’e0’ (u’,i) x 1 ’/y 1 e1’e1’ x a ’/y a ea’ea’ x 0 = x 0 ’

22. EE392P Project: Time-Varying Encoders for Constrained Systems22 Sliding-Block Decoders The memory and/or anticipation of a sliding- block decoder can vary from phase to phase. Case of only two phases. Let x={x 0, x 1 }. If, then only one round of splitting will be needed because  states of phase 0 need not to be split;  for any edge starting in phase1, its terminal state is in phase 0 and so its weight is 1; it follows that there is an x-consistent splitting that reduces all weights of phase 1 to 1. The anticipation of the resulting decoder will be 1 in phase 0 and 0 in phase 1 in the finite-type case.

23. EE392P Project: Time-Varying Encoders for Constrained Systems23 Sliding-Block Decoders Proposition 5: Let x={x 0, x 1, …, x k-1 }. If, then at most k-1 rounds of splitting are required. In the finite-type case, in phase j, the sliding-block decoder will have anticipation at most k-j-1. Sliding block decoder window size In this case, the sliding-block decoder will stop looking ahead periodically. If the sliding-block decoder has no memory, the code is actually block-decodable on blocks of length.

24. EE392P Project: Time-Varying Encoders for Constrained Systems24 Conclusion The paper showed how to adapt the state- splitting algorithm to the time-varying setting. The framework of time-varying encoders is useful to design high-rate codes with reduced decoder error propagation and reduced complexity. PRML (4,4) constraint is an example of a constrained system whose design benefits a lot from the time-varying approach.

25. EE392P Project: Time-Varying Encoders for Constrained Systems25 References Main paper: J. J. Ashley, and B. H. Marcus, “Time-Varying Encoders for Constrained Systems: An approach to Limiting Error Propagation”, IEEE Trans. on Inform. Theory, vol. 46, May 2000. Other papers: R. L. Adler, J. Friedman, B. Kitchens, and B. H. Marcus, “State splitting for variable-length graphs,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 108-113, Jan. 1986. B. H. Marcus, P. H. Siegel, and J. K. Wolf, “Finite-state modulation codes for data storage,” IEEE J. Select. Areas Commun., vol. 10, pp.5-37, Jan. 1992. C. D. Heegard, B. H. Marcus, and P. H. Siegel, “Variable-length state splitting with applications to average runlength-constrained (ARC) codes”, IEEE Trans. Inform. Theory, vol. 37, pp.759-777, May 1991.